Toward a Unified Methodology for Fractal Extension of Various Shape Preserving Spline Interpolants

Abstract

Fractal interpolation, one in the long tradition of those involving the interpolatary theory of functions, is concerned with interpolation of a data set with a function whose graph is a fractal or a self-referential set. The novelty of fractal interpolants lies in their ability to model a data set with either a smooth or a nonsmooth function depending on the problem at hand. To broaden their horizons, some special class of fractal interpolants are introduced and their shape preserving aspects are investigated recently in the literature. In the current article, we provide a unified approach for the fractal generalization of various traditional nonrecursive polynomial and rational splines. To this end, first we shall view polynomial/rational FIFs as α-fractal functions corresponding to the traditional nonrecursive splines. The elements of the iterated function system are identified befittingly so that the class of α-fractal function f α incorporates the geometric features such as positivity, monotonicity, and convexity in addition to the regularity inherent in the generating function f . This general theory in conjuction with shape preserving aspects of the traditional splines provides algorithms for the construction of shape preserving fractal interpolation functions. Even though the results obtained in this article are generally enough, we wish to apply it on a specific rational cubic spline with two free shape parameters.

Bicubic partially blended rational quartic surface

“…To construct α-fractal function f α , the procedure can be easily carried out, which is the content of the following proposition. Details may be consulted in [14]. Let us introduce the following notation for a continuous function g defined on a compact interval J : m(g; J) = min {g(x) : x ∈ J}, M (g; J) = max {g(x) : x ∈ J}.…”

“…Thus researchers keep trying to find best possible function that can interpolate the data with shape preserving property. As a submissive contribution to this goal, Chand and coworkers have initiated the study on shape preserving fractal interpolation and approximation using various families of polynomial and rational IFSs (see, for instance, [12,13,14,15,16,17,18,19]). These shape preserving fractal interpolation schemes possess the novelty that the interpolants inherit the shape property in question and at the same time the suitable derivatives of these interpolants own irregularity in finite or dense subsets of the interpolation interval.…”

Multivariate Affine Fractal Interpolation

“…Fractal functions are not well explored in the field of shape preserving interpolation/approximation. Motivated by theoretical and practical needs, the authors have initiated the study of shape preserving interpolation and approximation using fractal functions [22,23,24,25,26,27,28,29,30,31,32].…”